The force Facting on a charge q with mass m moving with velocity v in a magnetic

Question

The force Facting on a charge q with mass m moving with velocity v in a magnetic field B is given by the Lorentz formula: F= q(v × B). Imagine a particle moving in the (x))- plane in a region where there exists a uniform (i.e., constant) B-field oriented along the z direction. Assuming Newton's second law, F m(dv/d), applies in this case, show that the force and the velocity are perpendicular and that both have constant magnitude. 1. 2. Let r = xi + yj + zk and define r = Iri = Vr-r = (x2 + y2 + z2. Then f = r/ Evaluate the following (a) V2(%) (b) V(r2) (c) V(%)

Explanation / Answer

1. force F acting on particle with charge q with mass m moving with speed v in magnetic field B is given by

F = q(v x B)

given B is along z direction, B = |B|k

particel moves in xy plane, so, v = ai + bj m/s

now, F = q(v x B) = q((ai + bj)*|B|k) = q|B|(bi - aj)

and F . v = q|B|(bi - aj).(ai + bj) = q|B|(ab - ab) = 0

hence Force and velocity are perpendicular to each other as their dot product it zero

so when force is perpendicular to velocity, the magnitude of velocity will remain constant, and when magnitude of v remains constant, then magnitude of F remains constanty too

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